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The Modified Truncated SVD Method for Regularization in General Form

1992; Society for Industrial and Applied Mathematics; Volume: 13; Issue: 5 Linguagem: Inglês

10.1137/0913066

2168-3417

Per Christian Hansen, T. Sekii, Hiromoto Shibahashi,

Scientific Research and Discoveries

The truncated singular value decomposition (SVD) method is useful for solving the standard-form regularization problem: $\min ||{\bf x}||_2 $ subject to $\min ||A{\bf x} - {\bf b}||_2 $. This paper presents a modification of the truncated SVD method, which solves the more general problem: $\min ||L{\bf x}||_2 $ subject to $\min ||A{\bf x} - {\bf b}||_2 $, where L is a general matrix with full row rank. The extra work, associated with the introduction of the matrix L, is dominated by a QR-factorization of a matrix with dimensions smaller than those of L. In order to determine the optimal solution, it is often necessary to compute a sequence of regularized solutions, and it is shown how this can be accomplished with little extra computational effort. Finally, the new method is illustrated with an example from helioseismology.